Simple Events are Mutually Exclusive

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Let $\EE$ be an experiment.

Let $e_1$ and $e_2$ be distinct simple events in $\EE$.

Then $e_1$ and $e_2$ are mutually exclusive.


By definition of simple event:

\(\ds e_1\) \(=\) \(\ds \set {s_1}\)
\(\ds e_2\) \(=\) \(\ds \set {s_2}\)

for some elementary events $s_1$ and $s_2$ of $\EE$ such that $s_1 \ne s_2$.

It follows that:

\(\ds e_1 \cap e_2\) \(=\) \(\ds \set {s_1} \cap \set {s_2}\) Definition of $e_1$ and $e_2$
\(\ds \) \(=\) \(\ds \O\) Definition of Set Intersection

The result follows by definition of mutually exclusive events.